PROJECT SUMMARY: Over the past several decades, non-invasive functional magnetic resonance imaging (fMRI) has revolutionized the study of brain function and organization, enhancing scientific understanding of normal brain function, development, aging and disease. Yet leveraging the full potential of fMRI data remains challenging due to its massive size, complex dependence structure and noise. Analysis of individual subjects, which is needed for clinical care and the study of brain-behavior relationships, is particularly difficult due to high noise levels and typical short scan durations. Traditional analysis techniques were originally developed with computational feasibility in mind, rather than optimal efficiency and power. Today, statistical, computational and data advances provide opportunities for development of statistical methods with substantially improved accuracy for group and individual fMRI analysis. In particular, cortical-surface fMRI (csfMRI) data, an increasingly popular format in which the cortical gray matter is projected to a 2-dimensional manifold, offers two important benefits. First, geodesic distances along the cortical surface are a meaningful measure of dissimilarity in neuronal activation, unlike Euclidean distances in traditional volumetric fMRI data, making csfMRI optimal for use in spatial models. Second, csfMRI data achieves more accurate alignment of subjects' cortical areas, thus improving the precision of group studies and providing an opportunity to borrow strength across subjects. This project focuses on the development of computationally efficient Bayesian statistical methods for csfMRI data. We address two specific scientific objectives: (1) estimation of activation in the brain in response to a task or stimulus, and (2) identification of functional areas of the brain, which tend to activate together in the absence of a particular task. For (1), we propose a spatial Bayesian model that addresses the limitations of previously proposed models by (a) utilizing csfMRI data rather than volumetric fMRI, (b) employing recent developments in spatial statistics and Bayesian computation for accurate and efficient model estimation, (c) utilizing an efficient excursions set method to identify areas of activation based on the joint (rather than the marginal) posterior distribution, and (d) proposing an efficient and principled multi-subject analysis approach. We also propose several extensions to allow for spatial dependencies that are not stationary and isotropic. For (2), we propose a hierarchical Bayesian independent component analysis (ICA) model that borrows strength from the population through empirical priors, which are estimated from large, publicly available fMRI datasets. The use of empirical priors is very computationally advantageous. Finally, we combine this model with the proposed spatial Bayesian approach to task activation developed for Aim 1 by incorporating a spatial prior appropriate for csfMRI data into the hierarchical ICA model. We conduct simulation and reliability studies to validate the proposed methods and compare them with traditional approaches. We also apply the proposed methods to studies of autism spectrum disorder and amyotrophic lateral sclerosis or Lou Gehrig's disease.